im on the Route to the Core for some years now, and some months ago i got my first permadeath run (yay!)
Recently started over, freeing Raisu and finished stuff in Eridani.
Now in the next System i found my first Korolov Base.

After doing some E200 Freighter missions for them i got promoted and allowed to escort Antares II/V-class freighters.
Now everytime half-way the escort a random Charon frigate shows up (not the Kronosaurus) and just kills the freighter. In this stage of the game it is kind of impossible to get that Frigate down fast enough (or even get it down at all).

I even bought a Nami Launcher with KM500 Missiles, but it still needs some time to get that Frigate down.
And when i kill it after the mission failed (repeated that and killed multiple Charons) they sent the Kronosaurus after me, which just focuses me instead of any freighter around. Dont stand any chance against that @#$!.

What are your recommendations for that? Just back off and do Korolov Shipping later in the game (i suppose doing the Arena Story would grant sufficient money/equipment; dual particle beam seems fine for that.)

Also i never managed to reach the highest Korolov Rank. I remember there are multiple Bases in one playthrough, but im not sure. Is the ranking depending on the Korolov Mission count? Or on the value of the Mission rewards?

Regarding the frigates, I find that getting in between the frigate and the freighter and intercepting the weapons fire with your ship will help to keep the freighter alive for a bit longer. EI7000 is the choice of freighters to escort as it has good shields, a turbolaser and it's fairly fast.The KM500 missiles are good against the frigates, even better if you have the targeting ROM installed. As you suggest, I would come back a bit later with more powerful weapons. Do the first mission, which is fairly easy, then upgrade and return later. The frigates are tough, and the Kronosaurus has much better armor than standard frigates which is why it is tougher to kill.

Korolov ranks are related to Korolov xp. You need 3000 xp points to be promoted to Master or Legend. You will only be promoted to Legend if you haven't failed a mission. If one of your freighters gets killed you will be promoted to Master instead. The XP points are gained by completing missions and destroying frigates but I'm not sure of how many points go with each, others will know.

When dealing with frigates, you have it right. Just buy an initial stock of Stilettos and use them, and the frigates will go down long before they become a threat to the freighter. If they're not dying fast enough to satisfy you, either use the guaranteed Smartcannon or an omnidirectional turbolaser cannon to supplement your firepower, which has the added benefit of blocking any missiles it fires at you. The Kronosaurus can be killed in the same way, though you should have some repair items on hand for the sake of safety, as its armor is better than what you're likely to have at that point.

While it's kind of cheating, you can also lead it over to a friendly station and the friendly station will kill it for you. The AI doesn't yet understand how to keep its distance when the thing it's chasing flees to a well defended station.

Regarding the 'highest korolov rank', it's almost impossible to get unless you're actively trying to get it. Either you'll successfully escort about a dozen freighters in a row and be locked out of future missions for breaking the record or you'll kill enough frigates that the Kronosaurus will come after you and you'll be locked out of future missions for killing it. There's not any hidden, unique content that you're missing out on at very high ranks, though - most of the interesting things are in places most players will reach in a normal run.

The getting locked out was my biggest concern tho, i recognized this from previous playthroughs.
I will try to do the 12 missions in a row some time later this week, lets see if thats enough for the Legend Rank!

After finishing the first Korolov station via the destruction of the Charon base in their system i panicked a little if there will be another Station in any other system (despite Charon).
Found my second Korolov station some Systems later and flew some other escort missios for them, until they congratulated me to get 10 successful escort missions in a row.
I needed to self-destruct some times cause of one of this nasty charon frigates blowing a freighter up, but respawning didnt change the "success in a row" thing.
Service Record:
EI 100 x2
EI 200 x4
EI 7000 x1
Ant. II ------
Ant. V x2
Ant. I -------

(Yes, this are 9 Missions...someone hid the 10th somewhere i cant find it)

Now im the new Korolov-Legend (but it doesnt say so in the congratulations thing, needed to check via. Service Record)

Sidenotes:
- Didnt even need to kill the Kronosaurus.
- The 3000 XP thing doesnt seem to apply or at least every mission grants 300 XP

The Kronosaurus is sent after you for killing frigates. When you kill 5 of them, the kronosaurus will hunt you, no matter where you are in the game. It is not tied to you escorting freighters. So this can still happen, for example in Charon, there are usually some frigates out there.